Electronic Density- Algorithm for Nonadiabatic Couplings in Molecular Dynamics Simulations

M. TOMMASINI,1 V. CHERNYAK,2 S. MUKAMEL2 1Dipartimento di Chimica Industriale e Ingegneria Chimica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy 2Department of Chemistry, University of Rochester, P. O. RC Box 270216 Rochester, NY 14627-0216

Received 25 February 2001; accepted 2 March 2001

ABSTRACT: Closed expressions for nonadiabatic couplings are derived using the collective electronic oscillators (CEO) algorithm based on the time-dependent Hartree–Fock equations. Analytic derivatives allow the calculation of transition density matrices and potential surfaces at arbitrary nuclear geometries using a molecular dynamics trajectory that only requires a CEO calculation at a single configuration. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 00: 1–14, 2001 Key words:

trix between ground and excited states, is computed Introduction instead [5]. These matrices further represent col- lective oscillators or electronic normal modes that he simulation of photoinduced processes in describe the time evolution of the of T molecules requires the computation of elec- the system under the effects of an external perturba- tronic potential surfaces and nonadiabatic cou- tion. The CEO eigenvalue equation can be written as plings between them [1, 2]. The collective electronic Lξk = kξk.(1) oscillators approach (CEO) provides an inexpensive algorithm for computing adiabatic potential sur- L is the effective TDHF Liouville operator for the faces [3, 4]. This constitutes a major computational molecule, whose spectrum describes the electronic saving since a quantity which carries much less in- excitations of the system. The rep- formation than the many electron wavefunction of resentation of the oscillators ξk provides a clear the excited states, namely the transition density ma- characterization of molecular excitations in real space [6]. The eigenvectors ξk are normalized ac- 1 = ¯ † = ¯ Correspondence to: S. Mukamel; e-mail: mukamel@chem. cording to: (ξk, ξk) Tr(ρ[ξk , ξk]) 1, where ρ is rochester.edu. Contract grant sponsor: National Science Foundation and the 1The scalar product introduced here to normalize the eigen- Petroleum Research Fund, American Chemical Society. vectors ξk is discussed in Appendix A.

International Journal of Quantum Chemistry, Vol. 00, 1–14 (2001) © 2001 John Wiley & Sons, Inc. TOMMASINI, CHERNYAK, AND MUKAMEL the Hartree–Fock ground state density matrix. The has been rewritten in terms of the ground state plus x sum of the excitation energy k(R)andtheground the excitation energy εk = ε0 + k.Thematrixh ˆx state energy ε0(R) at different nuclear configura- represents the operator h in the chosen basis set tions R gives the excited state adiabatic potential ˆx = x † h nmhnmcncm. The expression for NAC involving surface εk(R). These quantities are sufficient to de- the ground state is even simpler and only requires scribe the nuclear dynamics of a molecule in regions † the knowledge of (ρi0)nm =i|cncm|0≡(ξi)nm and of the nuclear space far away from avoided cross- † † (ρ0i)nm =0|c cm|i≡(ξ )nm ings or conical intersections. However, many in- n i teresting photoinduced processes involve the tran- ξ x Tr(ξ †hx) x = Tr( ih ) x = i sition between different potential surfaces in the Ai0 ; A0i .(5) i i vicinity of an avoided crossing or conical intersec- The matrices {ξ } are obtained from the collective tion [7]. An example of such a process is the cis to i electronic oscillators algorithm [6, 12]. The transi- trans photoisomerization of rhodopsin which is the tion density matrices between excited states, ρ ,can elementary process in vision [8 – 10]. Under these ik be also expressed in terms of the CEO [13]. To use circumstances it is crucial to include the nonadia- ˆx batic couplings between different potential surfaces Eqs. (4) and (5) we need to express the operator h in order to properly describe the joint electronic and in terms of the derivative of the matrix elements of the molecular Hamiltonian. In Appendix C we nuclear dynamics, and to predict the branching ra- ˆ tios among different reaction pathways. show that the matrix elements of the operator hx The aim of this work is to derive closed ex- canbeexpressedinanatomicorbitalrepresenta- pressions for the nonadiabatic couplings (NAC) tion as between any pair of electronic states in the CEO hx = tx + Vxρ¯ (6) framework. The close connection between the NAC and the derivatives of the density matrices further where tx is the derivative of the matrix elements provides the formal solution of the equations pro- for the one-electron part of the Hamiltonian and Vx posed in [3] in terms of CEO eigenvectors and NAC. is the derivative of the electron–electron interaction ˆ The first-order nonadiabatic couplings are con- operator. The operator hx represents the vibronic veniently defined in terms of a vector potential in coupling of the molecular Hamiltonian and it is the nuclear space [11] source of the nonadiabatic coupling. These results       will be discussed in the coming sections. The sec-  ∂  Ax ≡− i k (2) ond section presents the derivatives of the ground ik ∂x state density matrix and of the CEO transition den- where i| and k| are two adiabatic electronic states sity matrices. In the third section we present the and x is a generic nuclear coordinate. In Appendix B equations for classical molecular dynamics on ex- we use perturbation theory to express the NAC in cited state hypersurfaces using the CEO/TDHF ap- the form proach. An analytic expression for the derivatives  |ˆx|  of the density matrices is also derived. In the fourth x = i h k = Aik ; i k (3) section we discuss some general properties of nona- εi − εk diabatic couplings resulting from the translational ˆx where h is the derivative of the molecular Hamil- symmetry of the electronic Hamiltonian when ex- ˆ 2 tonian H with respect to a nuclear coordinate x. pressed using atomic orbitals centered on the nuclei. This expression is convenient for numerical compu- Similar properties are derived also for the density tations. Introducing a basis set, with the correspond- matrices. In the fifth section we discuss the role † ing fermionic creation and annihilation operators cn of nonadiabatic couplings in determining a contri- and cn,wehave bution to vibrational infrared absorption intensi- ρ x ties. Finally, the Appendixes present details of the x = Tr( ikh ) = Aik ; i k.(4)derivations and the CEO algorithm. i − k =| † |  (ρik)nm i cncm k denotes the one-electron tran- sition density matrix and the excited state energy Analytic Derivatives

2Throughout this work we use the short-hand notation and denote the derivative with respect to a nuclear coordinate x by x By solving the Hartree–Fock equation for the superscript. density matrix at various nuclear configurations

2 VOL. 00, NO. 0 NONADIABATIC COUPLINGS   and the CEO eigenvalue problem [Eq. (1)] it is possi- x = ξ , Lxξ ; i  i i      bletoconstructtheentireadiabaticpotentialenergy Lxζ = F(ρ¯)x, ζ + Vxζ , ρ¯ + Vζ , ρ¯x . surfaces. The nonadiabatic couplings at each nu- clear configuration can then be computed from the Here and in the remainder of this paper ζ denotes a knowledge of the CEO transition matrices and the matrix representing an arbitrary single electron op- transition energies, using Eqs. (4) and (5). A differ- erator: Eq. (9) shows the action of superoperators in ent strategy for achieving the same goal is to use Liouville space. The scalar product (ζ1, ζ2)inEq.(9) the analytic derivatives of all quantities with re- is defined as   spect to the nuclear coordinates. It is then possible † (ζ , ζ ) = (ζ |ζ ) = Tr ρ¯[ζ , ζ ] . (10) in principle to carry out the full Hartree–Fock/CEO 1 2 1 2 1 2 calculation only once, at some chosen initial config- With respect to this scalar product the Liouville op- uration. The analytic derivatives then allow us to erator Lζ = [F(ρ¯), ζ ]+[Vζ , ρ¯]ofEq.(1)isHermitian: x compute the quantities of interest at all configura- (Lη1, η2) = (η1, Lη2) (see Appendix A). F(ρ¯) is the tions by running a trajectory; we can thus generate derivative of the Fock matrix corresponding to the the density matrices on the fly while avoiding a new ground state density matrix ρ¯ (see Appendix H). Hartree–Fock SCF or a new CEO computation in the displaced nuclear configuration. Such strategy is reminiscent of Car–Parrinello simulations [14]. In Classical Nuclear this section we provide closed expressions for these Dynamics Simulations derivatives. Numerical considerations should deter- mine which strategy is to be preferred: repeating the Many processes and spectroscopic properties CEO or using the derivatives. Making use of these may be adequately simulated by running classical results for the NAC it is possible to compute the trajectories on excited state surfaces. However, the derivative of the ground state density matrix and classical description breaks down at curve crossing the derivative of the CEO eigenvectors. points where a quantum wave packet calculation is A general relationship exists between the deriv- required [7]. In [3] the Hartree–Fock equation for ative of the density matrices and the NAC (see Ap- the ground state and the CEO algorithm for the pendix D)   TDHF were differentiated with respect to nuclear coordinates in order to get closed equations that can ρx = Ax ρ − ρ Ax .(7) ij ik kj ik kj describe the molecular dynamics on excited state k = i k = j hypersurfaces. We recast here the equations of [3] In particular we have for the ground state density using a somewhat different notation and adopting matrix  a different point of view which underscores the ¯x = x − x † interesting role of the nonadiabatic couplings. We ρ A0jξj Aj0ξj .(8) j>0 consider both electronic and nuclear degrees of free- dom and write equations of motion for the variables Equation (8) shows that the derivative of the ground of interest along the classical trajectory of the nuclei. state density matrix is given by a weighted sum The state of the molecule along this trajectory at a over the CEO modes, where the coefficients are given time τ will be defined by the following quan- given by the nonadiabatic couplings between the tities: the nuclear coordinates x(τ), the Hartree–Fock ground and the various excited states. These nona- ground state density matrix ρ¯(x(τ)), and the CEO diabatic couplings can be computed analytically transition matrix ξi(x(τ)). On a given ith excited state using Eq. (5) once the CEO eigenvectors are known surface these satisfy the equations of motion: by solving Eq. (1). A closed expression may also be   derived for the derivative of the transition density ¨ d mαxα =− E0(x) + i(x) , matrix ξi (see the third section) dxα ∂ρ¯   x   xα ˙ x ξk(ξk, Di ) x = ρ¯ xα ,(11) ξ = + (1 − 2ρ¯) ξi, ρ¯ ; ∂τ i − + α = k i k>0; k<0; k i ∂ξ     i = xα ˙ x = x ¯ ¯ ξi xα . Di Ci , ρ , ρ ; ∂τ       α Cx =−hx, ξ − Vxξ , ρ¯ − Vρ¯x, ξ (9) i  i  i i The ground state density matrix ρ¯ satisfies for − ¯x + x Vξi, ρ i ξi; each nuclear configuration x(τ) the self consistent

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 3 TOMMASINI, CHERNYAK, AND MUKAMEL

Hartree–Fock equation [F, ρ¯] = 0, where F is the It is then possible to compute ρ¯x once the NAC in- Fock operator which depends on ρ. In the classical volving the ground state are obtained through the equation of motion for the nuclear coordinates [the knowledge of the CEO eigenmodes and the matrix ˆ first of Eqs. (11)] we used the sum of the ground elements of hx [Eq. (6)]. On the other hand, using state energy and the excitation energy to generate Eq. (15) and the algebraic properties of the CEO the excited state surface eigenvectors, we can express the NAC involving the   x 1 ground state as scalar products of p with the CEO Ei(x) = E0(x)+i(x) = 2Tr ρ¯t+ ρ¯Vρ¯ +i(x). (12) 2 eigenvectors:   The excitation energy i at each nuclear configura- x = ¯x A0j ξj, ρ ; tion x is the eigenvalue of the CEO equation, Eq. (1).   (16) x = † ¯x The necessary excited state energy gradient can be Aj0 ξj , ρ . computed as This alternative to Eq. (5) for computing the NAC     d xα 1 xα is particularly useful when the derivative of the E (x) +  (x) = 2Tr ρ¯t + ρ¯V ρ¯ x 0 i 2 density matrix is already available (e.g., ρ¯ may be dxα   xα easily obtained from a finite differences Hartree– + ξi, L ξi . (13) Fock SCF computation). Here the ground state energy gradient has been ex- We next turn to the derivative of the CEO eigen- pressed using the derivatives of the one-electron x = modes ξi . Applying ∂/∂x to the equation Lξi iξi parameters txα and two-electron parameters Vxα . gives [3] The derivative of the Liouville operator with re- (L −  )ξ x =−(Lx − x)ξ . (17) spect to a nuclear coordinate Lxα can be expanded as i i i i the last of Eqs. (9). Equations (11) together with the The right-hand side of Eq. (17) is assumed to be x x analytic expressions for the derivatives of the den- known and will be denoted Ci . Ci can be directly sity matrices along the nuclear coordinates, e.g., ρ¯xα computed once the equation for ρ¯x, Eq. (14), has xα and ξi , Eqs. (8) and (9), are closed. been solved and the derivative of the CEO eigen- x = x These equations can be solved by computing the value i (ξi, L ξi) has been evaluated. More CEO spectrum at a given initial point and then by explicitly, expanding the derivative of the Liouville propagating the density matrices along the classical operator5 we get trajectory.3 This follows the spirit of Car–Parrinello (L −  )ξ x = Cx; simulations on the ground state [14]. i i  i     x =− x ξ − xξ ρ¯ − ρ¯x ξ We next show how to solve the two equations for Ci h , i  V i, V , i (18) − ¯x + x the derivatives of the density matrices in terms of Vξi, ρ i ξi. the CEO eigenvectors and the NAC. In Appendix C we show that it is possible to expand The derivative of the ground state density matrix the derivative of the ground state density matrix in satisfies the following linear equation (see Appen- the CEO eigenmodes ξi. The reason is that the deriv- dix C, Eq. (72) and Appendix F):   ative of the density matrix belongs to the particle– Lρ¯x = ρ¯, hx . (14) hole space and the eigenmodes of L form a complete x basis set for that space (see Appendix E). On the Using the perturbative expression for ρ¯ , Eq. (79), x other hand ξi does not belong to the particle–hole and the expressions for the NAC involving the space spanned by the set of the CEO eigenvectors ground state, Eq. (5), we can express the solution of {ξi}. This is because the particle–hole spaces associ- Eq. (14) as follows:4 ¯  ated with different density matrices ρ are in general ¯x = x − x † different. Nevertheless, if the two density matrices ρ A0jξj Aj0ξj . (15) j>0 are close enough (as when evaluating the derivative x of ξi) it is possible to express the part of ξi lying ¯ 3For example, if we consider a given small displacement δ of outside the particle–hole space associated with ρ,in a given nuclear coordinate x, we can propagate the density ma- terms of the derivative of the density matrix ρ¯x (Ap- trix ρ¯ from x to x+δ through the use of the analytic derivative ρ¯x, pendix G). ξ x may thus be expressed as a sum of an ¯ + ≈¯ +¯x i e.g., ρ(x δ) ρ(x) ρ (x)δ. interband γ x and an intraband ϕx component 4The Hermiticity of ρ¯x is assured by the following property i i x ∗ =− x x ∗ =− x x = x + x of the NAC: (A0j) Aj0;(Aj0) A0j.Thispropertycan ξi γi ϕi . (19) be directly derived from the perturbative expression of the NAC, ˆ Eq.(3).Notethathx is Hermitian. 5See, e.g., the last of Eqs. (9) and Appendix H.

4 VOL. 00, NO. 0 NONADIABATIC COUPLINGS

These two components are evaluated in Appen- way. The integrals over atomic basis functions that x dix G. ϕi is obtained by making use of Eq. (108): enter the definition of the Hamiltonian in the atomic x = − ¯ ¯x + x ϕi (1 2ρ)[ξi, ρ ] . γi is given by Eq. (114). orbitalbasissetmustdependonlyontherelative Thus we finally obtain the analytic expression of the positions of the atoms, thus tij and (ij|kl)arein- derivative of the CEO transition density matrix (the variant to an overall translation of the molecule. prime denotes that the ith mode is excluded from Through the same reasoning used for εk we imme- the sum) diately get    x   ξ (ξ , D ) xλ xλ x = k k i + − ¯ ¯x t = 0; (ij|kl) = 0. (27) ξi (1 2ρ) ξi, ρ +, (20) ij k − i λ λ k>0; k<0 where This yields the following sum rules for the deriv-    atives of the Coulomb and exchange operator, the x = x ¯ ¯ 6 x Di Ci , ρ , ρ . (21) Fock matrix and h    Jxλρ¯ = Kxλρ¯ = 0; Vxλρ¯ = 0; Sum Rules For Nonadiabatic Couplings λ λ λ (28) and Transition Density Matrices Fxλ = 0; hxλ = 0. λ λ We first consider a sum rule for Hellman– In the atomic orbital representation the NAC Eq. (4) Feynman forces. The invariance of the electronic assumes the form energy εk(R) with respect to a rigid translation T x (ρik)nm(h )nm of nuclear coordinates {R}, results in the following Ax = nm : i = k. (29) ik  −  sum rule: i k    If we express the sum rule for hx in the atomic ˆxλ  = k(R) h (R) k(R) 0, (22) orbital representation, Eq. (28), we obtain the fol- λ lowing sum rule for the NAC: ˆxλ ˆ    wherewehavedefinedh (R) ≡ ∂H(R)/∂x . yλ λ Axλ = A = Azλ = 0. (30) This simply implies that the sum of the Hellman– ik ik ik λ λ λ Feynman forces acting on the nuclei is zero, both for the ground or for an excited state. The sum is ex- This relationship can be used as a consistency check tended over all N atoms, and the same equation can for the computation of NAC. be written for the y and the z derivatives. A rigid Let us turn now to the derivative of the one- translation T acting on the nuclear space cannot particle transition density matrix connecting two |  |  change the electronic energy, i.e., εk(R + T) = εk(R). electronic levels i and j in the atomic orbital rep- x ≡  | † |  In particular for an infinitesimal translation  along resentation (ρij)nm (∂/∂x) i cncm j . Combining the the x axis, we have above sum rule for the NAC with Eq. (7) we imme- diately get the following sum rule for the transition ε (R + ) = ε (R) (23) k k density matrices: ε +   expanding k(R )tofirstorderin ,resultsin N N N  xλ = yλ = zλ = ∂εk ρij ρij ρij 0. (31) εk(R) +  (R) = εk(R) (24) ∂ λ = 1 λ = 1 λ = 1 λ xλ As special cases of Eq. (31) we have for the deriva- which immediately gives  tives of the ground state density matrix and the CEO ∂εk eigenvectors {ξ } (R) = 0. (25) i ∂x λ λ N N N ¯xλ = ¯yλ = ¯zλ = Making use of the Hellmann–Feynman theorem ρ ρ ρ 0; [see Appendix B, Eq. (66)], we obtain λ = 1 λ = 1 λ = 1 N N N (32)     xλ yλ zλ ∂εk ˆ  ξ = ξ = ξ = 0. (R) = k(R) hxλ (R) k(R) = 0. (26) i i i ∂ λ = 1 λ = 1 λ = 1 λ xλ λ

Sum rules for nonadiabatic couplings and den- 6The sum rule for the Fock matrix holds provided that a simi- sity matrix derivatives can be derived in a similar lar sum rule exist for the density matrix. This is introduced later.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 5 TOMMASINI, CHERNYAK, AND MUKAMEL

We further note that the first-order nonadiabatic Discussion couplings may be used to compute also the second- order nonadiabatic couplings. This can be done as Infrared intensities may be related to the electron follows: mobility within molecules. They also enter the ex-          ∂2   ∂ ∂  pressions for vibrational hyperpolarizabilities [16]. xy ≡   =  ˆ  Aij i j i 1 j . (36) The contribution of the electronic charge to the ∂x∂y ∂x ∂y molecular dipole moment can be written in terms ˆ By considering the resolution of the identity 1 = ρ¯ of the ground state density matrix and the dipole |kk|,wehave operator µs (the superscript s denotes the sth Carte- k             sian component) xy  ∂   ∂  A = i k k j . (37)   ij ∂x ∂y Ms = Tr µsρ¯ . (33) k This gives The infrared intensity is quadratic in the derivatives  xy y of the dipole moment with respect to nuclear coor- A = Ax A . (38) dinates [17]. The derivative of Ms with respect to a ij ik kj k generic nuclear coordinate x is

∂M ∂µs ∂ρ¯ = Tr ρ¯ + Tr µs . (34) Appendix A: The CEO Algorithm ∂x ∂x ∂x The first term is known as the charge contribution We introduce here the basic quantities required to the infrared intensity. It represents the contribu- for the CEO algorithm. The molecular Hamiltonian tion to the change in the dipole moment arising can be written using second quantization as follows from the displacement along the vibrational coordi- [20, 21]:   nate x of fixed charges associated with the atomic ˆ † 1 † † H = tijc cj + ij|klc c clck orbital basis set. The second term, known as the flux i 2 i j contribution, reflects the redistribution of electronic ij ijkl   charges upon nuclear motion. This term can be di- − Es s † (t) µijci cj (39) rectly related to the nonadiabatic coupling vector s = x,y,z ij x potential between the ground and excited states A0j. † To show this we recall the expansion of the density ci and ci are the creation and annihilation oper- matrix derivative in the CEO eigenmodes, Eq. (15), ators associated with the atomic spin orbital χi. 7 x x x † These orbitals are assumed to be orthonormal, i.e., i.e., ρ¯ = A ξj − A ξ . Substitution of Eq. (15) j>0 0j j0 j χ |χ =δ . These operators satisfy the Fermi anti- in Eq. (34) gives i j ij  commutation relationships ∂Ms ∂µs           = ¯ + x s − x s † † † † Tr ρ A0j Tr µ ξj Aj0 Tr µ ξ . c , c = δ ; c , c = 0; c , c = 0. ∂x ∂x j i j + ij i j + i j + j>0 (40) (35) Here the matrix t represents the one-particle part The infrared flux is thus given by a sum of the of the electronic Hamiltonian, and the two-electron transition dipoles between the ground state and the integrals are denoted by ij|kl≡(ik|jl). The single- manifold of excited states weighted by the nonadia- electron matrix elements tij, the two-electron matrix batic couplings between ground and excited states. elements (ij|kl), and the dipole ele- For conjugated molecules, such as Carotenoids, in ments µ ij may be expressed in terms of the atomic which the ECC infrared bands are dominated by contributions only from few excited states, the ex- 7Usually atomic orbitals basis sets are not orthonormal, i.e., perimentally accessible infrared intensities [18, 19] ≡¯|¯ = Sij χi χj δij. The algebra is less involved if an orthonor- could be used to estimate the magnitude of the mal basis set is used, so that it is convenient to transform the {¯} { } nonadiabatic couplings. Making use of the NAC nonorthonormal basis set χi into an orthonormal one χi ,as sum rule it is possible to show that the sum of the follows: − |χ =|¯χ  1/2 infrared fluxes with respect to the Cartesian coor- i j Sji ; dinate xλ, yλ,orzλ must vanish. This is consistent − χ |=S 1/2¯χ |; with the charge conservation principle from which i ij j − − − − χ |χ = 1/2¯χ |¯χ  1/2 = 1/2 1/2 = δ the sum rule on infrared fluxes is usually derived. i j Sik k l Slj (S )ikSkl(S )lj ij.

6 VOL. 00, NO. 0 NONADIABATIC COUPLINGS orbitals χ (r)as Let us consider now the spectrum of the L opera- i    tor (the frequencies k must be real to be physically = ∗ −1∇2 + tij χ (r) v(r) χj(r) dr; meaningful in a system without dissipation) i 2 r  = Z Lξk kξk. (50) v(r) =− I ; |r − RI| I Taking the Hermitian conjugate of this equation and  ∗ ∗ (41) χ (r1)χj(r1)χ (r2)χl(r2) making use of the anti-Hermiticity of L we get (ij|kl) = i k dr dr ; 1 2 † † † † |r1 − r2| † = ⇒ =−  L ξk kξk Lξk kξk . (51)  = ∗ µij χi (r)rχj(r) dr. The eigenvectors then come in conjugate pairs, † − ξk and ξk , with eigenvalues k and k,respec- Invoking the Hartree–Fock approximation, the one- tively. We introduce the following notation whereby ¯ = electron ground state density matrix (ρ)nm positive and negative indexes refer to the positive  | † |  0 cncm 0 satisfies the equation    and negative part of the spectrum ( k is taken to be F(q¯), ρ¯ = 0, (42) positive when k is positive): † → − ≡ → − ≡− ∀ where the Fock operator F is defined as follows: ξk k ξ k ξk  k k k > 0. (52) F(ρ) = t + Vρ. (43) The formal solution of Eq. (48) can be written as The action of the operator V on the density matrix = − takes into account the interaction among electrons. δρ(t) exp( iLt)δρ(0). (53) For a closed shell system the action of V on the The exponential of the Liouville operator works as a Hartree–Fock self-consistent ground state density propagator of the initial perturbation on the ground ¯ matrix ρ is (using the atomic orbital basis set): state density matrix δρ(0). Its spectral representation (Vρ¯) = 2( Jρ¯) − (Kρ¯) (44) is (ζ represents a generic one particle matrix) ij  ij ij  − − = ikt ( Jρ¯)ij = (ij|kl)ρ¯kl (45) exp( iLt)ζ ξke (ξk, ζ ). (54) kl k>0; k<0 (Kρ¯)ij = (ik|jl)ρ¯kl. (46) Acting with this propagator on δρ(0) we get  kl δρ(t) = z (t)ξ ; J and K are commonly known as the Coulomb and k k k>0; k<0 the exchange operators, respectively. −i t (55) zk(t) =zk(0)e k ; The CEO equations may be derived by starting   = with the time-dependent Hartree–Fock equation zk(0) ξk, δρ(0) .   ρ˙(t) =−i F(ρ), ρ . (47) A scalar product (η1, η2) of two one-electron density matrices in the Liouville space spanned by L can be The Fock matrix F is time-dependent through ρ(t). defined as Assuming a small time-dependent deviation from    ≡ | = ¯ † the equilibrium ground state density matrix, ρ(t) = (η1, η2) (η1 η2) Tr ρ η1, η2 . (56) ¯+ ρ δρ(t), it is possible to derive directly from Eq. (47) This is an unusual scalar product since it is anti- the following linearized equation, where terms of Hermitian, i.e., (η , η ) =−(η†, η†). With respect δρ2 1 2 2 1 order have been neglected, and Eq. (42) has been to this scalar product the Liouville operator L be- considered: comes Hermitian: (Lη1, η2) = (η1, Lη2). This can be δρ˙ =−iLδρ. (48) explained as follows. Let us consider two vectors η and ζ belonging to the Liouville space spanned by The operator Lζ = [F(ρ¯), ζ ] + [Vζ , ρ¯]isdefined the CEO eigenvectors given by Eq. (50). We can over the space of single particle density matrices writeforthescalarproduct(η, ζ ) the following ex- and represents an effective Liouville operator for the pansion over the CEO eigenmodes: Hartree–Fock Hamiltonian. The Hermiticity of the  † = density matrix δρ δρ implies that its time deriva- (η, ζ ) = (η, ξk)(ξk, ζ ). (57) = ˙ tive ρ(t) δρ(t) must be Hermitian at all times and k that the Liouville operator must be anti-Hermitian Similarly the action of the Liouville operator L on ˙ † = ˙ ⇒ † =− ⇒ † =− = (δρ) δρ iL δρ iLδρ L L. (49) a generic vector ϕ k ξk(ξk, ϕ)canberepre-

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 7 TOMMASINI, CHERNYAK, AND MUKAMEL sented as matrix δρ(t) has to be a particle–hole matrix (the  proof follows from Appendix E if a δρ =¯ρxδx is Lϕ =  ξ (ξ , ϕ). (58) k k k considered). As a direct consequence the eigenvec- k tors of the Liouville operator L are also particle–hole Making use of the above two expressions, and as- matrices, see Eq. (55). suming ζ = Lϕ, we can express (η, Lϕ)as  = (η, Lϕ) (η, ξk)(ξk, ξh)h(ξh, ϕ). (59) Appendix B: TDHF Calculation k,h of Nonadiabatic Couplings Using the anti-Hermicity of the scalar product † † The NAC can be expressed in a form suitable (η1, η2) =−(η , η ), we obtain 2 1 for the TDHF applications making use of perturba- =− † † † † † † (η, Lϕ) (ϕ , ξh )(ξh , ξk )k(ξk , η ). (60) tion theory. According to the definition of the NAC, k,h Eq. (2), we need to express the first derivative of Note that because of the orthogonality of the modes, the wavefunction with respect to the nuclear co- (ξ , ξ ) = δ , we are allowed to change  into  ordinate x. To that end we perturb the molecular k h kh k k  when going from Eq. (59) to Eq. (60). Making use of Hamiltonian by introducing a small displacement  the notation with negative indexes, Eq. (52), we get for a given nuclear coordinate x.Tofirstorderin  we have † † (η, Lϕ) = (ϕ , ξ− )(ξ− , ξ− )− (ξ− , η ). (61) ˆ   h h k k k ˆ ˆ ∂H H(x + ) = H(x) +  + o 2 . (64) k,h ∂x The sum over the CEO modes runs over both posi- We can apply standard perturbation theory [22] to tive and negatives indexes; Eq. (61) is equivalent to get the first-order approximations to the eigenval-  + | +  = † † ues εk(x ) and eigenvectors k(x ) of the (η, Lϕ) (ϕ , ξh)(ξh, ξk)k(ξk, η ). (62) ˆ perturbed (displaced) Hamiltonian H(x+). For the k,h eigenvalues we have The right-hand term of Eq. (62) is identified with     † †  ˆ  (ϕ , Lη ) through the use of Eq. (59), and we finally ∂H  2 εk(x + ) = εk(x) +  k(x) k(x) + o( ). (65) have ∂x (η, Lϕ) = (ϕ†, Lη†) =−(L†η, ϕ) = (Lη, ϕ). (63) This can be immediately used to calculate the deriv- ative of the eigenvalue itself The second equality holds because of the anti- ∂ε (x) ε (x + ) − ε (x) Hermicity of the scalar product and the last one k = lim k k ∂ →0  results from the anti-Hermicity of L. x      ˆ  The CEO algorithm is aimed at computing a ∂H  = k(x) k(x) . (66) few low-energy eigenvalues and the corresponding ∂x eigenvectors of the operator L [Eq. (50)]. The compu- Equation (66) is the celebrated Hellmann–Feynman tation is carried out with the Lanczos type algorithm theorem for the computation of the atomic forces in discussed in [12]. According to Eq. (55) the CEO molecules [23]. The same line of arguments can be eigenvectors can be interpreted as electronic normal used to calculate the derivative of the wavefunction modes which represent the time evolution of the den- as well. The first-order correction to the wavefunc- sity matrix following an initial deviation from the tion is   equilibrium δρ(0). Another useful way to view the   k(x + ) = k(x) matrices ξk comes from the expression of the linear   ∂ ˆ response of the system in terms of ξk [13]. By com-    H     j(x) ∂x k(x) 2 paring the usual sum over states expression with the −  j(x) + o  . (67) ε (x) − ε (x) > = j k CEO expression involving the ξk, ξk can be identi- j 0; j k fied with the one-electron transition density matrix The derivative is thus given by between the ground and the excited state of en- † ∂|k(x) |k(x + )−|k(x) ergy εk = k + ε0, i.e., (ξk)nm =k|cncm|0.Itis = lim ∂ →0  worth noting that if we express the density matrix x      ∂ ˆ   j(x) H k(x) (68) as a single Slater determinant at all times (TDHF =−  ∂x j(x) − . approximation), ρ(t) becomes idempotent. Thus the j>0; j = k εj(x) εk(x)

8 VOL. 00, NO. 0 NONADIABATIC COUPLINGS

According to Eq. (2) we need to compute the deriv- elements of the Hamiltonian with respect to an ative of the full many electron excited state wave- arbitrary nuclear displacement x. To that end we function to get the NAG. We shall show that this consider the perturbation on the Hartree–Fock solu- can be done with a considerably reduced informa- tion for the ground state induced by a displacement tion (i.e., the one-particle transition density matrices along x. Applying the derivative operator ∂/∂x to obtained from a TDHF computation). Substituting the Hartree–Fock equation [F(ρ¯), ρ¯] = 0 we get (see the expression for the derivative of the wavefunc- Appendix H for the explicit definition of the deriva- tion Eq. (68) into the definition of the NAG, Eq. (2), tives of the various terms) we get ∂     [t + Vρ¯, ρ¯] = tx + Vxρ¯ + Vρ¯x, ρ¯ + t + Vρ¯, ρ¯x  |ˆx|  ∂ x = i h k = x       Aik − ; i k, (69) = ρ¯ ρ¯x + ρ¯x ρ¯ + x + xρ¯ ρ¯ εi εk F( ),  V ,  t V , ˆ ˆ = ρ¯x − ρ¯ x + xρ¯ = where we have introduced the notation hx ≡ ∂H/∂x. L , t V 0. (71) It is important to note that this is a one-electron The same Liouville operator L that describes elec- operator. Indeed, when the derivative of the mole- tronic excitations induced by an external electric cular Hamiltonian is evaluated with respect to an field [6, 15], also represents the perturbation on the arbitrary nuclear coordinate x, the two-electron part density matrix induced by a small displacement of of the Hamiltonian involving the electron–electron nuclear coordinates: repulsion does not depend on the position of the nu-   ˆ ρ¯x = ρ¯ x + xρ¯ clei, and does not enter hx.Whenanatomicorbital L , t V . (72) basis set is employed, we have vibronic coupling The effect of this displacement is equivalent to also in the coefficients of the two-electron part of the adding an external field  that perturbs the isolated  |  ˆ Hamiltonian (39), i.e., ij kl depends on the nuclear molecule Hamiltonian (64) through the operator hx ˆx positions. This does not imply that h is a two- instead of the dipole operator µˆ coupled to the elec- ˆ particle operator, but the matrix elements of hx in tric field E. This allows us to identify the matrix ˆ an atomic orbital representation will depend also on elements of hx with tx + Vxρ¯.8 This can be put on ˆ the derivative of the two-electrons integrals. hx can a firm basis as shown below. The derivative of the † x thus be written in terms of the creation (cn)andan- idempotency condition of ρ¯ implies that ρ¯ belongs nihilation (cm) operators introduced earlier to the particle–hole space (see Appendix E), so that ˆ  it may be expanded in terms of the eigenvectors of L ∂H ˆ  ≡ hx = hx c†c . (70) ∂x nm n m ρ¯x = axξ + bxξ †. (73) nm j j j j j>0 To calculate the matrix elements of this opera- tor entering Eq. (69) we can thus use the TDHF These eigenvectors are orthonormal and they come = † one-particle transition density matrices (ρij)nm in conjugate pairs ξj, ξj with eigenvalues j  | † |  − i cncm j , given in [13] in terms of the eigenvectors ξi and j, respectively. Using the scalar product that of the Liouville operator L. Introducing ρij and the makes the Liouville operator Hermitian, Eq. (56), transition energies j = εj − ε0 into Eq. (69) we get we have the following orthonormality relations for Eq. (4). The expression for NAC which involve the a given CEO eigenvector:   ground state, Eq. (5), is simpler and merely requires = ¯ † = † (ξ, ξ) Tr ρ[ξ , ξ] 1; the knowledge of ρi0 ≡ ξi and ρ0i ≡ ξ ,whichare   i † † † available from a CEO computation. (ξ , ξ ) = Tr ρ¯[ξ, ξ ] =−1;   (74) Thus TDHF computation of the NAC requires the (ξ, ξ †) = Tr ρ¯[ξ †, ξ †] = 0; explicit expression for the matrix elements of the   ˆ (ξ †, ξ) = Tr ρ¯[ξ, ξ] = 0. one-electron operator hx, Eq. (70). This will be de- x x rived in Appendix C. To compute the coefficients aj and bj of Eq. (73), we consider the scalar product of Eq. (72) with the ˆ Appendix C: The Expression for hx 8The equation for the first-order perturbation ξ(t) to the ground state density matrix ρ¯ with respect to an electric field We can obtain a closed expression for the op- E − =−E ¯ ˆ (t)is:i[∂ξ(t)/∂t] Lξ(t) (t)[µ, ρ] [6]. In the limit of a static erator hx in terms of the derivative of the matrix field E this reduces to Lξ = E[ρ¯, µ].

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 9 TOMMASINI, CHERNYAK, AND MUKAMEL

† eigenvectors ξj and ξ   j    Appendix D: Derivative of the ¯x = ¯ x + x ¯ ξj, Lρ ξj, ρ, t V ρ ; Single-Electron Ground State      (75) † ¯x = † ¯ x + x ¯ Density Matrix ξj , Lρ ξj , ρ, t V ρ . Substituting the expansion of ρ¯x into Eqs. (75), mak- Using the perturbative expression for the deriv- ing use of the orthonormality conditions (74) for the ative of the wavefunction, we can immediately get left-hand side of Eqs. (75) we get a closed expression for the derivative of the single-    x x x † electron density matrix. Considering the wavefunc- ξj, Lρ¯ = ξi, L a ξi + b ξ i i i tion approximated by a single Slater determinant, i>0  we can apply the standard definition of the one- = x − x † ξj, ai iξi bi iξi electron density matrix in the second quantized > ¯ = | † |  i 0 form: (ρ)nm 0 cncm 0 . The derivative of the den- = x = x aj j(ξj, ξj) jaj , (76) sity matrix can be carried out and ∂ ¯x =  | † |     (ρ )nm 0 cncm 0 ξ †, Lρ¯x = ξ †, L axξ + bxξ † ∂x j i i i i i ∂ | ∂|  = 0 † | + | † 0 i>0 c cm 0 0 c cm . (81)  ∂x n n ∂x = ξ †, ax ξ − bx ξ † j i i i i i i To evaluate this expression we need the deriva- i>0  tive of the ground state bra and ket. Making use of =− x † † = x bj j ξj , ξj jbj . (77) Eq. (68), we have This yields the following expression for the coeffi-  ˆ ∂|0 j|hx|0 cients: =− |j ; x x ∂x  (ξ ,[ρ¯, t + V ρ¯]) > j x = j j 0 (82) aj ;  |   |ˆx|  j ∂ 0 =− 0 h j  | † ¯ x + x ¯ (78) j . (ξ ,[ρ, t V ρ]) ∂x j x = j j>0 bj . j We have denoted the transition energy from the Since both ρ¯ and tx + Vxρ¯ are Hermitian, Eq. (78) ground state as j = εj−ε0. Substituting Eq. (82) into shows that the coefficients, like the eigenmodes, Eq. (81) we obtain for the derivative of the ground come in conjugate pairs, bx = (ax)†. This guaran- j j state density matrix tees that the derivative of the density matrix is also Hermitian. As must be the case because ρ¯x is the   |ˆx|   |ˆx|  ¯x =− 0 h j  | † | + | † |  j h 0 derivative of the ρ¯.Perturbative (ρ )nm j cncm 0 0 cncm j . j j calculation gives for the derivative of the density j>0 matrix (see Appendix D) (83) Finally adopting the definition of the CEO eigen-  ξ x ξ † + ξ † x ξ † † † Tr( jh ) j Tr( j h ) j modes (ξ ) =j|c c |0,(ξ ) =0|c c |j,andthe ρ¯x =− . (79) j nm n m j nm n m ˆx x † j = j>0 second quantized version of h nm hnmcncm,we get ρ¯x is thus expanded over the basis set of the CEO †  † x x † transition density matrices ξj, ξ .Comparingthe Tr(ξ h )ξj + Tr(ξjh )ξ j ¯x =− j j result of the expansion carried out using the coeffi- ρ . (84) j x x j>0 cients aj and bj . [Eq. (78)] with Eq. (79) we can make the following identifications: Note that the derivative of the density matrix turns      ˆ x → ¯ x + x ¯ =− † x out to be Hermitian: (ρ¯x)† =¯ρx (the operator hx aj ξj, ρ, t V ρ Tr ξj h ;      (80) is Hermitian). This is the direct consequence of the bx → ξ †, ρ¯, tx + Vxρ¯ =−Tr ξ hx . j j j Hermiticity of the density matrix. Using the same Finally, using the definition of the scalar product approach it is possible to derive closed expressions (ζ1, ζ2) and expanding the left-hand term of the for the derivative of excited states density matrices above equalities, we obtain Eq. (6) (see Appendix F). and transition density matrices over excited states.

10 VOL. 00, NO. 0 NONADIABATIC COUPLINGS

x The final result is The equation for bj is redundant and it is simply  ˆ  ˆ x   i|hx|k k|hx|j the conjugate of the equation for aj . Equation (89) x = − x x x ρij (ρkj)nm (ρik)nm . (85) finally gives the expression for h = t + V ρ¯. εi − εk εk − εj k = i k = j Let us represent our matrices in the molecular By substituting the NAG from Eq. (69) in the above orbital basis set expression for ρx, we obtain Eq. (7). This shows that ij 0 α 0 β† ξ = ; ξ † = ; it is possible to expand the derivative of the general j β 0 j α† 0 transition density matrix in terms of transition den- (90) 00 AB sity matrices (vectors in Liouville space) where the ρ¯ = tx + Vxρ¯ = . coefficients are given by the NAG. 01 CD Expanding the scalar product in Eqs. (80) using its Appendix E: Representation of ρ¯x definition, Eq. (56), we have to compute the trace in the CEO Basis Set of a nested pair of commutators multiplied by the density matrix        The derivative of the idempotency condition † ξ , ρ¯, tx + Vxρ¯ ≡ Tr ρ¯ ξ , ρ¯, tx + Vxρ¯ . (91) ρ¯2 =¯ρ implies that the derivative of the density j j matrix ρ¯x belongs to the space spanned by the CEO We shall compute the right-hand side of Eq. (91) eigenvectors (the particle–hole subspace of the sin- step by step. The inner commutator can be ex- gle Slater determinant space). To show that we start panded as follows: with the derivative of the idempotency

  ρ¯xρ¯ +¯ρρ¯x =¯ρx 0 −B . (86) ρ¯, tx + Vxρ¯ = . (92) C 0 We adopt the representation for the density matrices9 Making use of the above expression, the outer com- mutator gives 00 x ab ρ¯ = ; ρ¯ = . (87)   01 cd † † † 0 −B β C + Bα 0 ξ , = . Expanding the matrix products in Eq. (86) using this j C 0 0 −α†B − Cβ† representation we get (93)

0 b ab Multiplying by ρ¯ and taking the trace we finally ob- = . (88) c 2d cd tain     This immediately implies that the blocks a and d Tr ρ¯ ξ †, ρ¯, tx + Vxρ¯ must be zero. The structure of ρ¯x is then block-off j  00 β†C + Bα† 0 diagonal, the same as the CEO eigenvectors. = Tr 01 0 −α†B − Cβ† − Tr(α†B) − Tr(Cβ†). (94) Appendix F: Identification of hx It is easy to show that except for a minus sign the In order to identify the matrix elements of hx we same result can be obtained from the following sim- have to consider Eqs. (80) and expand the scalar pler formula: product on the left-hand side. Below we show that   x x    † this scalar product (ξ ,[ρ¯, t + V ρ¯]) is given by † 0 β AB j Tr ξ tx + Vxρ¯ = Tr (95) − { † x + x ¯ } j α† 0 CD Tr ξj (t V ρ) , so that Eqs. (80) reduce to       β†C β†D ax → ξ , ρ¯, tx + Vxρ¯ =−Tr ξ † tx + Vxρ¯ = Tr j j j α†A α†B =− † x † † Tr(ξj h ). (89) = Tr(Cβ ) + Tr(α B). (96)

9 In this representation the blocks are identified as follows: We have made use of the cyclic invariance of the UU UO . Occupied MO’s are labeled O and unoccupied MO’s = OU OO trace, Tr(AB) Tr(BA). This proves the first equality are labeled U. in Eq. (89).

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 11 TOMMASINI, CHERNYAK, AND MUKAMEL

Let us compute the inner commutator of Eq. (102) at Appendix G: Derivative of a General first order in δρ¯ and δξ Particle–Hole Matrix 0 β K + δK ≈ [ξ + δξ, ρ¯ + δρ¯] ≈ −α 0 To discuss the variation δξ of a general interband βδρ¯† − δρα¯ δβ matrix ξ induced by a change δρ¯ of the ground state + . (103) −δα αδρ¯ − δρ¯†β density matrix10 it is convenient to introduce a pro- jection operator P in the Liouville space spanned by Making use of Eq. (103) it is easy to compute the the CEO eigenvectors. P projects a general matrix two terms of the double commutator (102). For the into the particle–hole subspace (i.e., the off diagonal particle–hole component this merely results in an block involving the occupied–unoccupied molecu- identity with respect to Eq. (101): lar orbitals). P is given by 0 β + δβ   [K + δK, ρ¯] = = ξ + δγ. (104) α + δα Pζ ≡ [ζ , ρ¯], ρ¯ . (97) 0 The useful information comes from the particle– When P acts on an arbitrary matrix in the sin- particle/hole–hole component = ζUU ζUO gle electron space ζ it extracts the K + δK δρ¯ ≈ K δρ¯ ζOU ζOO [ , ] [ , ] particle–hole component βδρ¯† + δρα¯ 0 = = δϕ. (105) 0 −(δρβ¯ + αδρ¯) = 0 ζUO Pζ . (98) To cast Eq. (105) in a standard matrix form, we make ζOU 0 use of the following simple relationships: As must be the case for any projection operator, P is βδρ¯† 0 δρα¯ 0 idempotent: P2 = P. For a general interband ma- ξδρ¯ = ; δρξ¯ = . 0 αδρ¯ 0 δρ¯†β trix ξ the following condition must be true:   (106) ξ = ξ ξ ρ¯ ρ¯ = ξ P ,i.e.,[ , ], . (99) We further recall how the density matrix itself acts ¯ ¯ on the particle–particle/hole–hole space A change δρ in the density matrix ρ, is accompanied by a change in the associated particle–hole space. UU 0 00 ρ¯ = ; Let us consider the following displacements for the 0 OO 0 OO (107) density matrix and for the general ξ matrix in the UU 0 UU 0 (1 −¯ρ) = . occupied–unoccupied molecular orbital representa- 0 OO 00 tion: Making use of Eqs. (106), (107) into Eq. (105) we 00 0 δρ¯ get the final expression for the particle–particle/ ρ¯ + δρ¯ = + (100) δρ¯† 01 0 hole–hole component of the variation of a general 0 β interband matrix ξ ξ + δξ = ξ + (δγ + δϕ) = α 0 δϕ = (1 − 2ρ¯)(ξδρ¯ + δρξ¯ ) = (1 − 2ρ¯)[ξ, δρ¯]+ . (108) 0 δβ δϕ 0 + + 2 . (101) Equation (108) can be used to express the intraband δα 0 0 δϕ1 x component ϕi of the derivative of a CEO eigenvec- x x = The variation of the interband matrix ξ consists of tor ξi , e.g., ϕi δϕi/δx. This result is used in Eq. (20) the particle–hole component δγ and the component of the third section. outside the particle–hole space δϕ. It is possible to Let us consider now the equation for the deriv- express δϕ in terms of δρ¯. To that end suffice it to ative of the CEO eigenvector, Eq. (18). In order to ξ x write Eq. (99) using the displaced quantities solve this equation for i , we need to consider the   projection operator P already defined with Eq. (97). + ¯ + ¯ ¯ + ¯ = + x [ξ δξ, ρ δρ], ρ δρ ξ δξ. (102) Acting with P on ξi projects out the particle–hole x = x component: Pξi γi . It is worth noting that P and L share the same eigenvectors so that they commute.11

10A nuclear displacement δx can be considered to be the cause 11 Any eigenvector ξ of L is a particle–hole matrix, so that of the variation δρ¯. Pξ = ξ [15].

12 VOL. 00, NO. 0 NONADIABATIC COUPLINGS

The shifted Liouville operator L − i has the en- the derivative of the one-electron part and the elec- tire spectrum of eigenvalues shifted by −i,butthe tron repulsion term of the Hamiltonian. Applying eigenvectors are still the same as those of L.Thus the derivative ∂/∂x on the matrix elements of Fij = − also L i commutes with P. Keeping this in mind tij + (Vρ)ij we have we can now directly apply P to Eq. (18). The action     of P on the left-hand side gives Fx = tx + Vxρ + Vρx . (115)     ij ij ij ij P (L −  )ξ x = (L −  )P γ x + ϕx = (L −  )γ x. i i i i i i i We consider a generic density matrix, not necessar- (109) ily the self-consistent Hartree–Fock one, ρ¯.Forthe The action of P on the right-hand side gives    one-electron matrix elements we have PCx ≡ Dx = Cx, ρ¯ , ρ¯ . (110) i i i ∂tij (tx) = . (116) Thus the projection of Eq. (18) in the particle–hole ij ∂x space is The derivative of the two-electron term is given by − x = x (L i)γi Di . (111)       Vxρ = 2 Jxρ − Kxρ This projection can be now expressed in terms of  ij  ij ij the eigenmodes ξi of the Liouville operator. Solv- xρ = | xρ J ij (ij kl) kl ing Eq. (111) requires inverting the Liouville oper- (117)   kl ator L.Ifthisoperatorisrepresentedintermsofits xρ = | xρ K ij (ik jl) kl spectrum, Lζ = ξkk(ξk, ζ ), the inversion k>0; k<0 kl becomes straightforward. The shifted Liouville op- erator in Eq. (18) can be written as and        − = − x x x (L i)ζ ξk[k i](ξk, ζ ) (112) Vρ = 2 Jρ − Kρ ij  ij ij k>0; k<0   x = | x Jρ (ij kl)ρkl and its inverse is ij (118) kl  ξ ξ ζ   − −1 = k( k, ) ρx = | ρx (L i) ζ , (113) K ij (ik jl) kl. k − i k>0; k<0 kl where the prime denotes that the ith mode is ex- We have defined the derivative of the two-electron cluded from the sum. This eliminates divergences integral x and is consistent with the fact that the derivative ξi , and so γ x, has no projection on ξ [3]. This definition ∂(ij|kl) i i (ij|kl)x ≡ . (119) of the inverse in a space orthogonal to the subspace ∂x spanned by the vector ξi alone can thus be safely x For a basis set independent on the nuclear co- used to compute γi . This is obtained directly by us- x = − −1 x ordinates (e.g., plane waves commonly used in ing this inverse, so that γi (L i) Di .Theresult is the following final expression: density functional calculations), the only source of vibronic coupling comes from the one-electron  ξ (ξ , Dx) x = k k i matrix elements t through the potential integral γi . (114)  ij k − i ∗ k>0; k<0 χi (r)v(r)χj(r) dr. In standard quantum chemistry Eq. (114) is used in Eq. (20) of the third section to computations a geometry-dependent atomic orbital compute the CEO eigenvector derivative, ξ x. basis set is commonly used, so that the vibronic cou- i plings enter also the two-electron matrix elements (ij|kl). This is the reason why the one-electron oper- ˆ Appendix H: Derivative of the ator hx described in the fifth section contains Vx as Fock Matrix well. Finally we note that the derivatives tx and (ij|kl)x Computing the derivative of the Fock matrix F introduced insofar are expressed in the orthonormal with respect to a nuclear coordinate x is required atomic orbital basis set. When a nonorthonormal for deriving the closed expression for the opera- basis is used, the relationships between the atomic ˆ tor hx (see Appendix C). This computation involves matrix elements in the two basis sets can be directly

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 13 TOMMASINI, CHERNYAK, AND MUKAMEL used (see Note 7) 5. Deumens, E.; Diz, A.; Longo, R.; Ohrn, Y. Rev Mod Phys 1994, 66, 917. = |ˆ| = −1/2˜ |ˆ|˜ −1/2 tij χi t χj Sik χk t χl Slj 6. Tretiak, S.; Chernyak, V.; Mukamel, S. J Am Chem Soc 1997, −1/2˜ −1/2 119, 11408–11419. = S tklS ; (120) ik lj 7. Ben-Num, M.; Martinez, T. J. J Chem Phys 1998, 108, 7244;  | = −1/2 −1/2 |  −1/2 −1/2 Ben-Num, M.; Martinez, T. J. Chem Phys 2000, 259, 237. ij kl Sii Sjj i j k l Skk Sll . 8. Wald, G. Science 1968, 162, 230. A relationship between the derivatives of the ma- 9. Schulten, K.; Humphrey, W.; Logunov, I.; Sheves, M.; Xu, D. trix elements in the two basis sets can be obtained Isr J Chem 1995, 35, 447. x  | x 10. Molnar, F.; Ben-Nun, M.; Martinez, T. J.; et al. J Mol Struct by differentiating these equations. tij and ij kl will involve the derivative of the overlap matrix S;for (THEOCHEM) 2000, 506, 169; Ben-Nun, M.; Molnar, F.; Lu, H.; et al. Faraday Discuss 1998, 110, 447. example for the one-electron matrix elements we 11. Chernyak, V.; Mukamel, S. J Chem Phys 2000, 112, 3572– have     3579. − x − − − − − x tx = S 1/2 tS˜ 1/2 + S 1/2t˜xS 1/2 + S 1/2t˜ S 1/2 . 12. Chernyak,V.;Schulz,M.F.;Mukamel,S.;Tretiak,S.;Tsiper, (121) E. V. J Chem Phys 2000, 113, 36–43. 13. Tretiak, S.; Chernyak, V.; Mukamel, S. Int J Quantum Chem 1998, 70, 711–727. 14. Car, R.; Parrinello, M. Phys Rev Lett 1985, 55, 2471; for a ACKNOWLEDGMENTS review see Remler, D. K.; Madden, P. A. Mol Phys 1990, 70, 921. The support of the National Science Foundation 15. Chernyak, V.; Mukamel, S. J Chem Phys 1996, 104, 444–459. and the Petroleum Research Fund sponsored by the 16.Castiglioni,C.;Tommasini,M.;DelZoppo,M.JMolStruct American Chemical Society are gratefully acknowl- 2000, 521, 137–155. edged. We wish to thank Dr. M. Ottonelli for many 17. Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; fruitful discussions. McGraw-Hill: New York, 1955. 18. Del Zoppo, M.; Castiglioni, C.; Zuliani, P.; Zerbi, G. In Handbook of Conducting Polymers; Skotheim, T. A.; Elsen- baumer, R. L.; Reynolds, J., Eds.; Dekker: New York, 1998. References 19. Gussoni, M.; Castiglioni, C.; Ramos, M. N.; Rui, M.; Zerbi, G. J Mol Struct 1990, 224, 445. 1. Domcke, W.; Stock, G. Adv Chem Phys 1991, 100, 1–169. 20. Avery, J. Creation and Annihilation Operators; McGraw- 2. Lengsfield III, B. H.; Yarkony, D. R. Adv Chem Phys 1992, Hill: New York, 1976. 82, 1–71. 21. Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: 3. Tsiper, E. V.; Chernyak, V.; Tretiak, S.; Mukamel, S. J Chem Introduction to Advanced Electronic Structure Theory; Phys 1999, 110, 8328–8337. McGraw-Hill: New York, 1989. 4. Tommasini, M.; Chernyak, V.; Mukamel, S. In Ultrafast Phe- 22. Pauling, L.; Wilson, E. B. Introduction to Quantum Mechan- nomena XII; Elsasser, T.; Mukamel, S.; Murnane, M.; Scherer, ics; McGraw-Hill: New York, 1935. N., Eds.; Springer-Verlag: Berlin, 2000. 23. Feynman, R. P. Phys Rev 1939, 56, 340.

14 VOL. 00, NO. 0